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Let $x$ be the space variable in which the differential equation is posed. Suppose we use Fourier transform to analysis these equations and let $\xi$ be the frequency variable. Then, $(x,\xi)$ is an element in cotangent bundle space which has symplectic structure.

When we solve a Hamiltonian Jacobi equation(or a first order non-linear equation) we need to use an analysis on the space variable $x$ and its 1-form $dx.$ The pair $(x,dx)$ which again is in cotangent bundle space. A good reference for an elementary treatment on this topic is Arnold's book.

Fourier Integral Operator is an operator which has Schwartz kernel as a distribution whose singularities are on a Lagrangian submanifold. Lagrangian submanifold is a topic of symplectic geometry.